Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

4. Graphing Trigonometric Functions

Graphs of Tangent and Cotangent Functions

Trigonometry

4. Graphing Trigonometric Functions

Graphs of Tangent and Cotangent Functions

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Problem 4.33

Textbook Question

Decide whether each statement is true or false. If false, explain why.
The tangent and secant functions are undefined for the same values.

Verified step by step guidance

Identify the definitions of the tangent and secant functions in terms of sine and cosine: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) and \( \sec(\theta) = \frac{1}{\cos(\theta)} \).

Determine when the tangent function is undefined: \( \tan(\theta) \) is undefined when \( \cos(\theta) = 0 \) because division by zero is undefined.

Determine when the secant function is undefined: \( \sec(\theta) \) is also undefined when \( \cos(\theta) = 0 \) for the same reason.

Identify the values of \( \theta \) where \( \cos(\theta) = 0 \). These occur at odd multiples of \( \frac{\pi}{2} \) (e.g., \( \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots \)).

Conclude that both tangent and secant functions are undefined for the same values of \( \theta \), specifically where \( \cos(\theta) = 0 \). Therefore, the statement is true.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Tangent Function

The tangent function, defined as the ratio of the sine to the cosine of an angle (tan(θ) = sin(θ)/cos(θ)), is undefined when the cosine of the angle is zero. This occurs at odd multiples of π/2 (90 degrees), where the function approaches infinity, leading to vertical asymptotes on the graph.

Secant Function

The secant function is the reciprocal of the cosine function (sec(θ) = 1/cos(θ)). It is undefined at the same angles where the cosine is zero, specifically at odd multiples of π/2 (90 degrees). Thus, secant also has vertical asymptotes at these points, indicating that the function does not have a defined value.

Undefined Functions

A function is considered undefined at certain points when it cannot produce a valid output. For both tangent and secant functions, this occurs at angles where the denominator of their respective ratios (cosine for tangent and secant) equals zero. Understanding these undefined points is crucial for analyzing the behavior of these trigonometric functions.